What is sequence? a basic definition . Advance definition for iit jam.

 Sequence is a enumerated collection of object in which repetition are allowed and order does matter.

Advance definition:- A sequence in a set S is a function , whose domain is the set N of natural numbers and whose range is a subset of S. 

Notes - " A sequence whose range is a subset of R is called a real sequence."

It  is usual to write a sequence as <an> or  like <a1, a2, a3, ......>

The number a1 ,a2 , a3, ..... are called the term or element of the sequence . Thus a1 is the first term , a2 is the second term ...an is the nth term of the sequence <an>.

Notes :- when the range set of a sequence is a subset of Complex number  that is  f:N-C   , then the function defines a sequence of complex numbers. 

A sequence can be described in several different ways ;

example:- < 1,4,9,16,25,36........> 

the formula for its nth term , < 1,4,9,16,25,36......n²..>  or as  < n²: nεN> or simply < n² > 

Most common types of Sequences:- 

1; Arithmetic Sequence 

2; Geometric Sequence

3; Harmonic Sequence

4; Recursive Sequence

5; Fibonacci Sequence

Range set of a Sequence;

The range set of a sequence is the set consisting of all distinct elements of a sequence and without regard to the position term . Thus the range may be finite or infinite set i.e. the range set of sequence <an > is given by { an : nεN}  or simply by  R(an)